TY - JOUR

T1 - A necessary condition for best approximation in monotone and sign-monotone norms

AU - Kimchi, E.

AU - Richter-Dyn, N.

PY - 1979/2

Y1 - 1979/2

N2 - Best approximation to f{hook} ε{lunate} C[a, b] by elements of an n-dimensional Tchebycheff space in monotone norms (norms defined on C[a, b] for which |f{hook}(x)| ≤ | g(x)|, a ≤ x ≤ b, implies ∥f{hook}∥ ≤ ∥g∥) is studied. It is proved that the error function has at least n zeroes in [a, b], counting twice interior zeroes with no change of sign. This result is best possible for monotone norms in general, and improves the one in [5]. The proof follows from the observation that, for any monotone norm, sgn f{hook}(x) = sgn g(x), a ≤ x ≤ b, implies ∥f{hook}- λg ∥ < ∥f{hook}∥ for λ > 0 small enough. This property is shown to characterize a class of norms wider than the class of monotone norms, namely "sign-monotone" norms defined by: |f{hook}(x)| ≤ |g(x)|, f{hook}(x) g(x) ≥ 0, a ≤ x ≤ b, implies ∥f{hook}∥ ≤ ∥g∥. It is noted that various results concerning approximation in monotone norms, are actually valid for approximation in sign-monotone norms.

AB - Best approximation to f{hook} ε{lunate} C[a, b] by elements of an n-dimensional Tchebycheff space in monotone norms (norms defined on C[a, b] for which |f{hook}(x)| ≤ | g(x)|, a ≤ x ≤ b, implies ∥f{hook}∥ ≤ ∥g∥) is studied. It is proved that the error function has at least n zeroes in [a, b], counting twice interior zeroes with no change of sign. This result is best possible for monotone norms in general, and improves the one in [5]. The proof follows from the observation that, for any monotone norm, sgn f{hook}(x) = sgn g(x), a ≤ x ≤ b, implies ∥f{hook}- λg ∥ < ∥f{hook}∥ for λ > 0 small enough. This property is shown to characterize a class of norms wider than the class of monotone norms, namely "sign-monotone" norms defined by: |f{hook}(x)| ≤ |g(x)|, f{hook}(x) g(x) ≥ 0, a ≤ x ≤ b, implies ∥f{hook}∥ ≤ ∥g∥. It is noted that various results concerning approximation in monotone norms, are actually valid for approximation in sign-monotone norms.

UR - http://www.scopus.com/inward/record.url?scp=49249146668&partnerID=8YFLogxK

U2 - 10.1016/0021-9045(79)90006-6

DO - 10.1016/0021-9045(79)90006-6

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AN - SCOPUS:49249146668

SN - 0021-9045

VL - 25

SP - 169

EP - 175

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 2

ER -